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Unit 12: The Moment Generating Functions

exists and is finite for all real numbers t belonging to a closed interval [–h, h]  , with Notes
h > 0, then we say that X possesses a moment generating function and the function
M : [–h, h]   defined by:
X
Mx(t) = E[exp(t(X)]
is called the moment generating function (or mgf) of X.
 Let X be a random variable possessing a moment generating function M (t). Define:
X
Y = a + bX

where a, b   are two constants and b  0.

12.6 Keywords

Moment generating function of a linear transformation: Let X be a random variable possessing
a moment generating function M (t). Define:
X
Y = a + bX
where a, b   are two constants and b  0.
Random variable: A random variable X is said to have a Chi-square distribution with n degrees
1
of freedom if its moment generating function is defined for any t  and it is equal to:
2
M (t) = (1 – 2t) -n/2
X
12.7 Self Assessment

1. If a random variable X possesses a moment generating function M (t), then, for any n ,
X
the n-th moment of X (denote it by  (n)) exists and is ..................
X
(a) random variables (b) same distribution
(c) b  0 (d) finite
2. Let X and Y be two .................. Denote by F (x) and F (y) their distribution functions and by
X Y
M (t) and M (t) their moment generating functions.
X Y
(a) random variables (b) same distribution
(c) b  0 (d) finite
3. X and Y have the .................. (i.e., F (x) = F (x) for any x) if and only if they have the same
X Y
moment generating functions (i.e. M (t) = M (t) for any t).
X Y
(a) random variables (b) same distribution
(c) b  0 (d) finite
4. Let X be a random variable possessing a moment generating function M (t). Define:
X
Y = a + bX
where a, b   are two constants and ..................
(a) random variables (b) same distribution
(c) b  0 (d) finite

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